@Chronos sketched one approach to the integral. It felt more natural to me in spherical coordinates, so here’s another version. If you know neither integral calculus nor spherical coordinates, I can answer differently.
Let \theta be latitude as measured from the north pole, with \theta=0 being the north pole and \theta=\pi being the south pole. For any latitude \theta, the velocity of interest is v_{\rm eq}\sin\theta, where v_{\rm eq} is the equatorial velocity. We want to average the velocity over all points on the sphere. How much each latitude contributes to that average is proportional to \sin\theta. Thus, the average is \left(\int_0^\pi(v_{\rm eq}\sin\theta)\sin\theta \,d\theta\right)/\left(\int_0^\pi\sin\theta \,d\theta\right), where the denominator normalizes our latitude weighting. The integral in the numerator evaluates to \frac{\pi}{2} times v_{\rm eq}, and the integral in the denominator evaluates to 2. Hence \frac{\pi}{4}v_{\rm eq}.